# Toward Variability Characterization and Statistic Models’ Constitution for the Prediction of Exponentially Graded Plates’ Static Response

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Functionally Graded Materials

_{c}denotes the volume fraction of the ceramic phase, and p denotes the exponent. Based on a selected distribution law, the material properties can be determined using Voigt’s rule of mixtures, among other possible homogenization schemes:

#### 2.2. Displacement Field and Constitutive Relation

#### 2.3. Uncertain Material and Geometrical Properties

**X**∼N(

**μ**,

**Σ**), i.e.,

**X**is a multivariate normal distribution with the mean value

**μ**and the covariance matrix

**Σ**. As one wants to ensure the independence between modeling parameters, the covariance matrix

**Σ**will be a diagonal matrix. To this purpose, taking into account the ability to ensure the independence between variables (Iman and Conover [29]), this study uses a Latin hypercube sampling (LHS) technique (Beck and Santos [30]).

#### 2.4. Multiple Linear Regression Model

_{i}

_{1}, x

_{i}

_{2}, …, x

_{in}), i = 1, 2, …, n, and the dependent variable, one has:

**0**,

**σ**). If the assumptions of this model are verified, a response prediction $\widehat{y}$ can be estimated via the least squares method from the sampled values.

^{2}^{2}) can be used. These coefficients are global statistics and provide a measure of the percentage of variability of the dependent variable that is explained by the model (for details, see Montgomery [31]).

## 3. Numerical Applications

#### 3.1. Verification Cases

_{m}= 70 GPa, ν

_{m}= 0.3) and zirconia (E

_{c}= 200 GPa, ν

_{c}= 0.3), yielding its graded mixture, an AlZrO

_{2}FGM. The static behavior of the FGM simply supported plate was compared with Bernardo et al. [22].

#### 3.2. Case Studies

#### 3.2.1. Uncertainty in the Material Properties and FGM Core Thickness

- Thinner Core (ec/h = 1/3)

_{m}and E

_{c}, have a higher contribution for the explanation of the transverse deflection ($w$) variability, being the most significant among all of the parameters. Table 4 presents the results of a new model (Equation (9)) with just the two most significant parameters, namely E

_{m}and E

_{c}.

^{2}, one can also observe a very good fitting.

- Thicker core (ec/h = 7/9)

^{2}= 0.9934, the independence assumption of the residuals is violated.

_{m}and E

_{c}, yielding a new model (LMR2), which is now presented in Equation (11). This LMR2 model obtained via a multiple linear regression model verified the residues’ assumptions of normality, null mean, constant variance, and the residues’ independence requisites.

_{m}and E

_{c}, is able to explain 97.68% of the maximum deflection variability. Again, the coefficients associated with each elasticity modulus reflected their different mean magnitude order.

#### 3.2.2. Uncertainty in the Material Properties and Whole Sandwich Thickness

- Thinner Core (ec/h = 1/3)

^{2}= 0.9921, the independence assumption of the residuals was violated. Therefore, a new model (LMR3) with just the elasticity modulus of the two materials, E

_{m}and E

_{c}, was built. This new model provided a good fit, Adj-R

^{2}= 0.9748, and the residual assumptions were verified. Table 8 presents the results of this model (Equation (12)) with just the two most significant parameters (E

_{m}and E

_{c}).

- Thicker Core (ec/h = 7/9)

_{m}and E

_{c}), another multiple linear regression model (LRM4) was built (Table 10), and the residuals’ assumptions of normality, null mean, constant variance, and independence were then verified.

#### 3.2.3. Verification of Statistic Models

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representation of the material phases mixture through the composite thickness, according to (

**a**) the power exponent law and (

**b**) exponential law.

**Figure 4.**Residual analysis for the model with all the inputs: case 3.2.1—Thinner core. (

**a**) Residual vs. fitted values; (

**b**) Normal QQ plot.

**Table 1.**Non-dimensional maximum deflection ${w}_{ndim}$ and deviation. Power law functionally graded materials (FGM).

a/h | Models | ${\mathit{w}}_{\mathit{n}\mathit{d}\mathit{i}\mathit{m}}$$\mathbf{(}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$ | $\mathit{D}\mathit{e}\mathit{v}(\%)$ |
---|---|---|---|

5 | Bernardo et al. [22] | 2.282 | - |

Present Q4 | 2.264 | 0.789 | |

Present Q9 | 2.280 | 0.088 | |

10 | Bernardo et al. [22] | 3.187 | - |

Present Q4 | 3.168 | 0.596 | |

Present Q9 | 3.194 | 0.220 | |

20 | Bernardo et al. [22] | 4.913 | - |

Present Q4 | 4.887 | 0.529 | |

Present Q9 | 4.920 | 0.142 |

a/h | ec/h | ${\mathit{w}}_{\mathit{n}\mathit{d}\mathit{i}\mathit{m}}$$\mathbf{(}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$ | Dev (%) | |
---|---|---|---|---|

Bernardo et al. [22] | Present Q9 | |||

5 | 1/3 | 2.874 | 2.874 | 0 |

2/3 | 2.834 | 2.834 | 0 | |

7/9 | 2.826 | 2.823 | 0.106 | |

10 | 1/3 | 4.082 | 4.074 | 0.196 |

2/3 | 4.003 | 3.995 | 0.199 | |

7/9 | 3.986 | 3.971 | 0.376 |

Properties | ec/h_{c} | E_{m} (GPa) | E_{c} (GPa) | ν_{m} | ν_{c} |
---|---|---|---|---|---|

μ (mean) | 1/3 | 70 | 200 | 0.3 | 0.3 |

LRM1 | Coefficients | 2.5% CI | 97.5% CI | p-Value |
---|---|---|---|---|

Intercept | 5.800 × 10^{−6} | 5.626148 × 10^{−6} | 5.974 × 10^{−6} | <2 × 10^{−16} *** |

Em | −2.157 × 10^{−17} | −2.332488 × 10^{−17} | −1.982 × 10^{−17} | <2 × 10^{−16} *** |

Ec | −7.021 × 10^{−18} | −7.635766 × 10^{−18} | −6.407 × 10^{−18} | <2 × 10^{−16} *** |

^{−8}on 27 degrees of freedom (DF), Multiple R

^{2}: 0.9778, Adj-R

^{2:}0.9762, F-statistic: 594.6 on 2 and 27 DF, p-value: <2.2 × 10

^{−16}. Significant codes: 0 ‘***’, 0.001 ‘**’, 0.01 ‘*’, 0.05 ‘.’, 0.1 ‘·’, 1.

Properties | ec/h | E_{m} (GPa) | E_{c} (GPa) | ν_{m} | ν_{c} |
---|---|---|---|---|---|

μ (mean) | 7/9 | 70 | 200 | 0.3 | 0.3 |

LRM2 | Coefficients | 2.5% CI | 97.5% CI | p-Value |
---|---|---|---|---|

Intercept | 5.677 × 10^{−6} | 5.501 × 10^{−6} | 5.834 × 10^{−6} | <2 × 10^{−16} *** |

Em | −2.003 × 10^{−17} | −2.171 × 10^{−17} | −1.835 × 10^{−17} | <2 × 10^{−16} *** |

Ec | −7.162 × 10^{−18} | −7.750 × 10^{−18} | −6.574 × 10^{−18} | <2 × 10^{−16} *** |

^{−8}on 27 degrees of freedom, Multiple R

^{2}: 0.9784, Adj-R

^{2}: 0.9768, F-statistic: 611.6 on 2 and 27 DF, p-value: <2.2 × 10

^{−16}. Significant codes: 0 ‘***’, 0.001 ‘**’, 0.01 ‘*’, 0.05 ‘.’, 0.1 ‘·’, 1.

Properties | ec/h | Cinf/h | Csup/h | E_{m} (GPa) | E_{c} (GPa) | ν_{m} | ν_{c} |
---|---|---|---|---|---|---|---|

μ (mean) | 1/3 | 1/3 | 1/3 | 70 | 200 | 0.3 | 0.3 |

LMR3 | Coefficients | 2.5% CI | 97.5% CI | p-Value |
---|---|---|---|---|

Intercept | 5.801 × 10^{−6} | 5.622 × 10^{−6} | 5.984 × 10^{−6} | <2 × 10^{−16} *** |

Em | −2.178 × 10^{−17} | −2.359 × 10^{−17} | −1.997 × 10^{−17} | <2 × 10^{−16} *** |

Ec | −6.958 × 10^{−18} | −7.591 × 10^{−18} | −6.326 × 10^{−18} | <2 × 10^{−16} *** |

^{−8}on 26 degrees of freedom, Multiple R

^{2}: 0.9765, Adj-R

^{2}: 0.9748, F-statistic: 560.8 on 2 and 27 DF, p-value: <2.2 × 10

^{−16}. Significant codes: 0 ‘***’, 0.001 ‘**’, 0.01 ‘*’, 0.05 ‘.’, 0.1 ‘·’, 1.

Properties | ec/h | Cinf/h | Csup/h | E_{m} (GPa) | E_{c} (GPa) | ν_{m} | ν_{c} |
---|---|---|---|---|---|---|---|

μ (mean) | 7/9 | 1/9 | 1/9 | 70 | 200 | 0.3 | 0.3 |

LMR4 | Coefficients | 2.5% CI | 97.5% CI | p-Value |
---|---|---|---|---|

Intercept | 5.661 × 10^{−6} | 5.479 × 10^{−6} | 5.844 × 10^{−6} | <2 × 10^{−16} *** |

Em | −2.024 × 10^{−17} | −2.209 × 10^{−17} | −1.840 × 10^{−17} | <2 × 10^{−16} *** |

Ec | −7.052 × 10^{−18} | −7.697 × 10^{−18} | −6.406 × 10^{−18} | <2 × 10^{−16} *** |

^{−8}on 27 degrees of freedom, Multiple R

^{2}: 0.974, Adj-R

^{2}: 0.972, F-statistic: 505.2 on 2 and 27 DF, p-value: <2.2 × 10

^{−16}. Significant codes: 0 ‘***’, 0.001 ‘**’, 0.01 ‘*’, 0.05 ‘.’, 0.1 ‘·’, 1.

ec | E_{m} | E_{c} | ν_{m} | ν_{c} | w(FEM) | w(LRM1) (Equation (10)) | $\mathit{D}\mathit{e}\mathit{v}$ $\mathbf{(}\mathbf{\%}\mathbf{)}$ |
---|---|---|---|---|---|---|---|

0.067 | 6.32 × 10^{10} | 1.83 × 10^{11} | 0.298 | 0.300 | 3.17 × 10^{−6} | 3.15 × 10^{−6} | −0.63 |

0.073 | 7.40 × 10^{10} | 1.96 × 10^{11} | 0.320 | 0.290 | 2.80 × 10^{−6} | 2.83 × 10^{−6} | 1.07 |

0.060 | 7.56 × 10^{10} | 1.92 × 10^{11} | 0.298 | 0.290 | 2.82 × 10^{−6} | 2.82 × 10^{−6} | 0.00 |

0.064 | 6.67 × 10^{10} | 2.19 × 10^{11} | 0.320 | 0.300 | 2.81 × 10^{−6} | 2.82 × 10^{−6} | 0.36 |

0.069 | 7.05 × 10^{10} | 2.10 × 10^{11} | 0.269 | 0.300 | 2.82 × 10^{−6} | 2.805 × 10^{−6} | −0.53 |

ec | E_{m} | E_{c} | ν_{m} | ν_{c} | w(FEM) | w(LRM2) (Equation (11)) | $\mathit{D}\mathit{e}\mathit{v}$ $\mathbf{(}\mathbf{\%}\mathbf{)}$ |
---|---|---|---|---|---|---|---|

0.162 | 6.35 × 10^{10} | 2.11 × 10^{11} | 0.317 | 0.300 | 2.87 × 10^{−6} | 2.89 × 10^{−6} | 0.70 |

0.141 | 6.68 × 10^{10} | 1.85 × 10^{11} | 0.297 | 0.300 | 3.02 × 10^{−6} | −2.66 × 10^{−6} | −1.88 |

0.171 | 7.05 × 10^{10} | 1.88 × 10^{11} | 0.280 | 0.290 | 2.91 × 10^{−6} | −2.76 × 10^{−6} | −1.95 |

0.156 | 7.67 × 10^{10} | 1.97 × 10^{11} | 0.326 | 0.300 | 2.70 × 10^{−6} | −2.95 × 10^{−6} | −2.09 |

0.162 | 6.35 × 10^{10} | 2.11 × 10^{11} | 0.319 | 0.300 | 2.87 × 10^{−6} | −2.783 × 10^{−6} | −1.97 |

ec | Cinf | Csup | E_{m} | E_{c} | ν_{m} | ν_{c} | w(FEM) | w(LRM3) (Equation (12)) | $\mathit{D}\mathit{e}\mathit{v}$ $\mathbf{(}\mathbf{\%}\mathbf{)}$ |
---|---|---|---|---|---|---|---|---|---|

0.070 | 0.0577 | 0.0636 | 7.1845 × 10^{10} | 1.7526 × 10^{11} | 0.285 | 0.285 | 3.036 × 10^{−6} | 3.017 × 10^{−6} | −0.64 |

0.066 | 0.0644 | 0.0686 | 6.6026 × 10^{10} | 2.1604 × 10^{11} | 0.289 | 0.327 | 2.855 × 10^{−6} | 2.860 × 10^{−6} | 0.17 |

0.071 | 0.0726 | 0.0695 | 7.2244 × 10^{10} | 2.3191 × 10^{11} | 0.316 | 0.322 | 2.613 × 10^{−6} | 2.614 × 10^{−6} | 0.06 |

0.068 | 0.0685 | 0.0757 | 7.4522 × 10^{10} | 2.0961 × 10^{11} | 0.317 | 0.296 | 2.708 × 10^{−6} | 2.719 × 10^{−6} | 0.41 |

0.065 | 0.0677 | 0.0680 | 6.3931 × 10^{10} | 1.7996 × 10^{11} | 0.271 | 0.320 | 3.184 × 10^{−6} | 3.157 × 10^{−6} | −0.87 |

ec | Cinf | Csup | E_{m} | E_{c} | ν_{m} | ν_{c} | w(FEM) | w(LRM4) (Equation (13)) | $\mathit{D}\mathit{e}\mathit{v}$ $\mathbf{(}\mathbf{\%}\mathbf{)}$ |
---|---|---|---|---|---|---|---|---|---|

0.163 | 0.0213 | 0.0210 | 6.6935 × 10^{10} | 2.0556 × 10^{11} | 0.297 | 0.299 | 2.845 × 10^{−6} | 2.857 × 10^{−6} | 0.42 |

0.161 | 0.0238 | 0.02528 | 7.4074 × 10^{10} | 1.9224 × 10^{11} | 0.307 | 0.332 | 2.770 × 10^{−6} | 2.806 × 10^{−6} | 1.30 |

0.157 | 0.0225 | 0.0195 | 6.6448 × 10^{10} | 2.2384 × 10^{11} | 0.324 | 0.303 | 2.723 × 10^{−6} | 2.738 × 10^{−6} | 0.54 |

0.149 | 0.0225 | 0.0216 | 7.2553 × 10^{10} | 1.7789 × 10^{11} | 0.320 | 0.310 | 2.927 × 10^{−6} | 2.938 × 10^{−6} | 0.39 |

0.175 | 0.0242 | 0.0231 | 7.3403 × 10^{10} | 2.0808 × 10^{11} | 0.321 | 0.295 | 2.691 × 10^{−6} | 2.708 × 10^{−6} | 0.62 |

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**MDPI and ACS Style**

Rosa, R.D.S.B.; Loja, M.A.R.; Carvalho, A.C.J.V.N.d.
Toward Variability Characterization and Statistic Models’ Constitution for the Prediction of Exponentially Graded Plates’ Static Response. *J. Compos. Sci.* **2018**, *2*, 59.
https://doi.org/10.3390/jcs2040059

**AMA Style**

Rosa RDSB, Loja MAR, Carvalho ACJVNd.
Toward Variability Characterization and Statistic Models’ Constitution for the Prediction of Exponentially Graded Plates’ Static Response. *Journal of Composites Science*. 2018; 2(4):59.
https://doi.org/10.3390/jcs2040059

**Chicago/Turabian Style**

Rosa, Rafael Da Silva Batista, Maria Amélia Ramos Loja, and Alda Cristina Jesus Valentim Nunes de Carvalho.
2018. "Toward Variability Characterization and Statistic Models’ Constitution for the Prediction of Exponentially Graded Plates’ Static Response" *Journal of Composites Science* 2, no. 4: 59.
https://doi.org/10.3390/jcs2040059